The Costs of Free Entry: An Empirical Study of Real Estate Agents in Greater Boston

Transcription

1 The Costs of Free Entry: An Empirical Study of Real Estate Agents in Greater Boston Panle Jia y Parag A. Pathak z This version: June 2011 Abstract This paper studies the real estate brokerage industry in Greater Boston, an industry with low entry barriers and substantial turnover. Using a comprehensive dataset of agents and transactions from , we nd that entry does not increase sales probabilities or reduce the time it takes for properties to sell, decreases the market share of experienced agents, and leads to a reduction in average service quality. These empirical patterns motivate an econometric model of the dynamic optimizing behavior of agents that serves as the foundation for simulating counterfactual market structures. A one-half reduction in the commission rate leads to a 73% increase in the number of houses each agent sells and bene ts consumers by about $2 billion. House price appreciation in the rst half of the 2000s accounts for 24% of overall entry and a 31% decline in the number of houses sold by each agent. Low cost programs that provide information about past agent performance have the potential to increase overall productivity and generate signi cant social savings. We thank Alan Genz for helpful suggestions in the early stage of this project. We thank Paul Asquith, Lanier Benkard, Steve Berry, Xiaohong Chen, JP Dube, Hanming Fang, Phil Haile, Brad Larsen, Ariel Pakes, Michael Powell, Tavneet Suri, Maisy Wong, Juanjuan Zhang, and seminar participants at Columbia, Harvard, John Hopkins, Michigan, Olin Business School, Toulouse, Wharton, and Yale for their comments. All errors are our own. Comments are welcome. y Department of Economics, MIT and NBER. pjia@mit.edu. Jia gratefully acknowledges the research support of the National Science Foundation via grant SES z Department of Economics, MIT and NBER. ppathak@mit.edu. 1

2 1 Introduction For a large majority of U.S. households, purchasing or selling a home is one of their most important nancial decisions. In 2007, two-thirds of households owned their homes, more than a quarter of national wealth was held in residential real estate, and there were 6.4 million sales of existing homes. 1 The National Association of Realtors (NAR) estimates that almost 80% of residential real estate transactions involve a realtor. Nationwide, brokers sales commissions exceeded $100 billion annually during the mid-2000s. 2 In Greater Boston, the site of our study, more than $4.1 billion in commissions was paid to realtors in the ten-year period from Several distinct features of the brokerage industry have long attracted popular and academic attention. First, the commissions that realtors charge are sizable, typically representing 5-6% of a property s transaction price. A property in our sample sells for $472,000 on average (in 2007 dollars). 3 At 5-6% of the sale price, the commission fee constitutes a signi cant transaction cost for most households, and is more than 40% of Massachusetts average median household income in Second, the commission rate displays little variation over time and across regions, and does not seem to re ect changes in the cost of selling houses. According to a recent federal government report (GAO 2005), commission rates within a market do not appear to vary signi cantly on the basis of the price of a home... and do not appear to have changed much in response to rapidly rising home prices in recent years. These facts have been described in several studies and contrast with the declining fees of intermediaries in other industries (see, e.g., Hsieh and Moretti (2003) and Levitt and Syverson (2008a)). With rising housing prices, a xed commission rate translates into a higher commission fee per transaction and makes working as a broker more lucrative. The absence of price competition with regards to commission fees is counter-intuitive given that there are low barriers to enter or exit the brokerage industry. Although all US states require licensing of brokers and salespersons, these requirements do not appear to present signi cant constraints. For instance, when the national average housing price increased by 83% from 1997 to 2006, NAR membership surged from 716,000 to 1,358,000 during the same period. In subsequent years, house prices fell slightly, but the total number of housing transactions plunged. Many agents stopped working as realtors and NAR reported a 20% reduction in its membership from 2006 to Similarly, entry peaked in 2004 in our Greater Boston sample when sales prices were highest, and dropped by about one-half by 2007 after prices declined. In this paper, we quantify the social costs of free entry and examine implications of the current xed commission structure. Our rich micro-level data set contains all properties in the Multiple Listing System (MLS) network from January 1998 to December 2007 in all cities and towns within a 15-mile radius of downtown Boston. There are 10,088 agents and 257,923 housing observations, each with detailed property attributes and transaction information. We are able to exploit cross-market 1 Sources for all numbers cited in this section can be found in the data appendix. 2 While a real-estate broker usually supervises an agent, often as the owner of the rm, and is subject to stricter licensing requirements, we use the terms agent, broker and salesperson interchangeably. 3 All dollar values in this paper are in terms of 2007 dollars, de ated using the urban CPI (series CUUR0100SA0). 2

3 variation across low and high-income suburbs of Boston and time-series variation from growing and declining local housing markets. The rst part of the paper documents that an increase in the number of agents neither improves the likelihood of sale nor reduces the amount of time required for properties to sell, but rather results in decreased market share for experienced agents. In addition, listings by inexperienced agents are 9% less likely to sell than listings by an agent with six or more years of experience. Having documented a strong business stealing e ect, we next formulate an econometric model to quantify the social costs of free entry, building on existing dynamic discrete choice models (see, e.g., Aguirregabiria and Nevo (2010)). Entry and exit decisions of agents, together with their observed commission revenue, allow us to identify the amount of income entrants could have alternatively earned had they not worked as agents. This foregone income is a measure of ine ciency since agents entry mostly dilutes the business of existing agents without increasing the total output of the brokerage industry. Our estimates imply that agents foregone income is about 80% of their observed revenue. By micro-founding the model on agent s choices, we then compute three counterfactuals taking into account agent re-optimization. First, motivated by FTC s investigations on rigid commissions, we present results from regulated across-the-board reductions in the commission rate. 4 Reducing the commission rate by one-half decreases entry by a third, increases the number of houses sold per agent by 73%, and raises the average sales likelihood by 2%. For our time period, this translates into $2 billion of savings for consumers and $900 million of savings in opportunity costs and entry costs from fewer real estate agents. The second counterfactual measures the alternative market structure if agents are compensated by their costs of selling properties, a potential outcome with free entry and exible commissions. Since these costs are unobserved, we treat the commissions that agents charged in 1998 as a conservative upper bound. If agents had been compensated by this upper bound, there would be 24% fewer agents and the number of houses each agent sold would increase by 31%. Total commissions would be reduced by $1.1 billion, and the social savings from fewer agents would be $525 million. Alternatively, this counterfactual can be interpreted as measuring how incentives to work as a realtor change with housing prices. Our estimates imply that only a small fraction of the bene ts from housing price appreciation is passed onto agents because of the business stealing from entrants. If there were no housing price appreciation during our sample period, the average commission would be $59,700; by comparison, when housing prices rose 1.5 times during the same period, the average commission only increased to $63,300. Finally, a xed commission rate makes it di cult for consumers to distinguish good agents from mediocre ones based on the prices that they charge. Our last counterfactual simulates the 4 The brokerage industry has been investigated for a number of reasons since at least the 1950s, including the case U.S. vs National Association of Real Estate Boards in 1950, which rst prohibited coordination of realtor fees. Recent FTC and Department of Justice investigations have also examined internet and virtual real estate o ces. See (last accessed February 2011) for additional information on the FTC s investigations of the real estate industry. 3

4 consequence of providing consumers with more information on the past performance of incumbent agents, a policy with low implementation costs given the records contained in the current MLS. By empowering consumers with more information, this policy reduces incentives for inexperienced agents to enter, resulting in an increased number of houses sold per agent and the potential for signi cant social savings. The remainder of the paper is structured as follows: Section 2 provides industry background, describes our data source, and reviews the related literature. Section 3 presents an initial empirical analysis of the real estate brokerage industry in Greater Boston. Section 4 develops our econometric model and Section 5 outlines the estimation approach. Section 6 describes our empirical results, while Section 7 presents the counterfactual analyses. The last section states our conclusions. Jia and Pathak (2011) (hereafter JP2) provides supplementary material on the sample construction, computational details, and additional results not reported here. 2 Industry Background and Data 2.1 Industry Background Real estate agents are licensed experts specializing in real estate transactions. They sell knowledge about local real estate markets and provide services associated with the purchase and sale of properties on a commission basis. For home sellers, agents are typically involved in advertising the house, suggesting listing prices, conducting open houses, and negotiating with buyers. For home buyers, agents search for houses that match their clients preferences, arrange visits to the listings, and negotiate with sellers. In addition, they sometimes provide suggestions on issues related to changes in property ownership, such as home inspections, obtaining mortgage loans, and nding real estate lawyers. All U.S. states require real estate brokers and agents to be licensed, but these requirements are minimal. In Massachusetts, applicants for a salesperson license need to take twenty-four hours of classroom instruction and pass a written exam. The quali cations for a broker s license involve a few additional requirements: one year of residence in Massachusetts, one year of active association with a real estate broker, completion of thirty classroom hours of instruction, passing a written exam, and paying a surety bond of ve thousand dollars. Agents, or salespersons, can perform most of the services provided by a broker, except that they cannot sell or buy properties without the consent of a broker. All licenses need to be renewed biennially, provided the license holder has received six to twelve hours of continuing education and has paid appropriate fees for renewal (currently, $93 for a salesperson and $127 for a broker). The general perception that these requirements are not signi cant deterrents to working as realtors is con rmed in our dataset where entrants account for about 13% of active agents each year. 4

5 2.2 Data The data for this study come from the MLS network for Greater Boston. We collected information on all listed non-rental residential properties for all towns within a fteen-mile radius of downtown Boston, with a total of 18,857 agents and 290,738 observations. 5 The list of 31 markets are shown in Figure 1, where we group together some towns and cities with few agents. The record for each listed property includes: listing details (the listing date and price, the listing rm and agent, commissions o ered to the buyer s agent, and so on), property characteristics, as well as transaction details (the sale price, date, the purchasing agent and rm) when a sale occurs. The number of days on the market is measured by the di erence between the listing date and the date the property is removed from the MLS database. We merge this data set with a database from the Massachusetts Board of Registration on agents license history which we use to measure their years of experience. Agents gender is provided by List Services Corporation, which links names to gender based on historical census tabulations. We exclude observations with missing cities or missing listing agents. Information on commissions charged by real-estate agents is di cult to obtain. Even though our data does not contain commissions paid to listing agents, it does contain commissions paid to buyer s agents. Jia and Pathak (2010) report that the buyer s agent commission is 2.0% or 2.5% for 85% of listings in the sample. Since we expect this to be a lower bound on commissions paid to the listing agent, in the analysis to follow, we assume that the total commission rate is 5% in all markets and years, and is split evenly between the seller s and buyer s agent. According to the 2007 NAR survey, most agents are compensated under a revenue sharing arrangement, with the median agent keeping 60% of his commissions and submitting 40% to his rm. We subsequently discuss how this assumption of a 60%-40% split impacts our analysis. The MLS dataset does not indicate whether working as a broker is an agent s primary occupation. To eliminate agents who may have brie y obtained access to the MLS system to buy and sell for themselves, we only keep agents who either work as a buyer s agent or a listing agent for more than 1.5 properties per year. This sample restriction leaves us with 10,088 agents listing 257,923 properties, about 90% of the original records. JP2 provides more details on the sample construction. Our analysis bene ts from three sources of variation present in the data: time-series variation in the housing market (from an up market to a down market), cross-sectional variation among agents (from green realtors to established agents with decades of experience), and geographical variation (the median household income of the most a uent town is more than three times higher than that of the poorest one). Table 1 shows that the number of listings varied from 20,000 to 23,000 in the late 1990s and early 2000s, but increased to 32,500 in There was a sharp decline in the number of listed houses in the following years when the housing market su ered from the 5 To verify MLS s coverage of transactions in the cities that we study, we compared it to the Warren Group s changes-of-ownership le based on town deeds records, which we have access to from This dataset is a comprehensive recording of all changes in property ownership in Massachusetts. The coverage was above 70% for all cities except Boston, which was around 50%. This fact, together with concerns about data quality in Boston, lead us to exclude the city of Boston from the empirical analysis. 5

6 decline in the aggregate economy. The weakness of the housing market in the latter part of the sample is apparent in the fraction of properties sold: before 2005, 75-80% of listed properties were sold; in 2007, only 50% were sold. The third column of Table 1 shows that average sales price was about $350,900 (in 2007 real dollars) in 1998 and peaked at $529,200 in It dropped slightly to $489,800 - $502,400 in 2006 and The amount of time it takes for properties to sell appears to lead the trend in sales prices: it sharply increases in 2005, and by the end of the sample a listed property requires about two months longer to sell than in Related Literature This paper is related to research on real estate brokers and their impact on the housing market. An important precursor to our work is Hsieh and Moretti (2003) which presents evidence consistent with socially ine cient numbers of real estate agents. Using variation across 282 metropolitan areas from the 1980 and 1990 Census of Population and Housing, they document that the average earnings of real estate agents are similar despite large di erences in housing prices, suggesting that agent entry dilutes rents associated with house price di erences under xed commissions. Moreover, agents in cities with high housing prices have lower productivity (measured by houses sold per agent) compared to agents in cities with low housing prices. Using the more recent ve percent sample of the 2000 Census of Population and Housing, Han and Hong (2011) examine agents variable costs of selling houses in a static entry model. Their estimates suggest that free entry leads to a loss of economies of scale: a 10% increase in the number of realtors increases the average variable cost of selling houses by 4.8%. Due to data limitations, they assume all agents are identical and ignore the opportunity cost of entry. Further a eld, there are a number of related papers on real estate agents. Kadiyali, Prince, and Simon (2009) study dual agency issues in real-estate transactions. Levitt and Syverson (2008b) compare home sales by agents who own the property to home sales by agents hired to sell the property. Hendel, Nevo, and Ortalo-Magné (2009) contrast traditional multiple-listing services with for-sale-by-owner platforms. Jia and Pathak (2010) study the impact of buyer s commissions on home sales. Aside from real-estate brokers, another paper which investigates the impact of free-entry is Berry and Waldfogel (1999) s study of the radio industry. Their paper is based on cross-sectional data on the number of stations, radio listening, and advertising prices. More recent contributions build dynamic econometric models of entry in imperfectly competitive industries (e.g., Collard- Wexler (2008), Dunne, Klimek, Roberts, and Xu (2009), Ryan (2010), and Xu (2008).) While sharing a methodological approach with these papers, our paper di ers from these studies on several dimensions. To capture the main features of the housing market while still allowing estimation to be feasible, we work with a model with six state variables. Rather than following the common approach of discretizing the state space, we treat it as continuous, approximate the value function using basis functions, and cast the Bellman equation as model constraints. We adopt a similar procedure in the counterfactual simulations. Our estimator falls into the class of estimators studied by Ai and Chen 6

7 (2003) and Chen and Pouzo (2009). Bajari, Chernozhukov, Hong, and Nekipelov (2009) provide identi cation results for dynamic games that apply to our application. In an independent study, Kristensen and Schjerning (2011) show that maximum likelihood estimation of dynamic models with value functions being approximated by basis functions has desirable properties. Finally, our paper is related to an extensive literature on dynamic models of occupational choice and job matching (see e.g., Keane and Wolpin (2009) for a recent survey). Related to our econometric approach, Keane and Wolpin (1994) develop a nite-period dynamic model and use backward induction to estimate the value function. In each step, they solve the value function for a subset of state points, and extrapolate to all other state points. 6 Our method may be applicable in similar problems with the advantage that it avoids discretization and is computationally much less demanding. 3 Initial empirical analysis 3.1 Descriptive statistics Greater Boston s housing market exhibits signi cant time-series variation in the number of properties listed, the likelihood of sale, and sales prices over our sample period. Total volume of home sales increased from 18,094 in 1999 to 21,432 in 2004, an increase of 18%. It subsequently decreased to 76% of the 1999 level within three years. The average real sales price of homes went from $385,900 in 1999 to $529,200 in 2004, down to $489,800 in Since the expected revenue of an agent depends on each of these factors, it is unsurprising that agent entry and exit follow these market-wide trends. Table 2A shows that the number of active agents increased from around 3,800 in 1998 to a peak of more than 5,700 in 2005, just as house price appreciated during the same period. The number of agents who left the industry was around during the early period, but rose to in the latter part of our sample, when housing market conditions deteriorated and the fraction of listed properties that were sold dropped. Agent performance is also related to overall trends in the housing market. During the run-up in prices in the early part of the 2000s, the number of properties each agent intermediated was about eight per year. By 2007, the average agent conducted a little more than ve transactions. The distribution of agents transactions is highly skewed: both the number of listings sold per agent and the number of houses bought per agent at the 75 th percentile is four to six times that of the 25 th percentile. During the down markets of 2006 and 2007, a signi cant fraction of real estate agents were hit hard: more than 25% of the listing agents did not sell any properties at all. Home sales, agent entry and exit, and agent performance also vary across markets within Greater Boston, as shown in Table 2B. The most expensive town in our sample is Wellesley, where the average house sold for more than $1 million. On the other end, in Randolph, the average sold price is $290,000. Quincy, a town with over 10,000 housing transactions, has signi cant turnover: 6 For a few other examples (including discussions) of value function approximation, see Ericson and Pakes (1995), Judd (1998), Farias, Saure, and Weintraub (2010), and Fowlie, Reguant, and Ryan (2011). 7

8 it is home to the most entries and has the second largest number of exits. Cambridge has about the same number of properties, but there are considerably fewer agents and much less turnover. This translates into a higher number of properties sold and bought per agent in Cambridge than in Quincy: versus In general, agents in higher-priced towns are involved in fewer transactions, and the correlation coe cient between the average housing price and the number of transaction per agent is around across all markets. An important component of performance di erences between agents is their experience. Panel A of Table 3 reports the average annual commissions of agents based on the number of years they worked as a broker. The category of nine or more years of experience has 19,210 observations with a total of 3,146 agents. These agents were active at the beginning of our sample. All other categories (one to eight years of experience) are mostly comprised of brokers who entered during our sample period. Agents who have worked for one year earn $20,000 on average. They sell about 61% of their listed houses and generate a larger share of their income from working as a buyer s agent. In contrast, agents with the most experience are 13% more likely to sell their listed properties, earn about $73,000 in commissions, and earn more of their commission income working as a seller s agent than as a buyer s agent. There is a clear monotonic pattern between measures of agent experience, sales probabilities, and commissions. Finally, more experienced agents appear to sell faster, although the di erence in days on market is modest. Performance di erences are also closely related to agent skill, which we measure by the number of transactions they brokered in the previous year. Panel B of Table 3 reports sales probabilities, days on the market, and commissions by deciles of agent skill. Since this measure is highly correlated with years of experience, Panel B displays similar patterns as Panel A. Agents in top deciles have higher sales probabilities, earn higher commissions, and a larger portion of their income comes from the selling side. Next we examine agents performance over time. We assign agents who were present in 1998 (the 1998 cohort) into four groups based on their 1998 commissions, and plot their annual commissions from 1999 to 2007 in Figure 2. Results using other cohorts are similar. The top quartile agents consistently earned $100,000 or more for most of the years, while the bottom quartile barely earned $30,000 in commissions even when housing prices were at their peak. Moreover, agents in the top quartile earn signi cantly more than those in the 2 nd or 3 rd quartile. The earning gap between the 2 nd and the 3 rd quartile agents is much smaller and also compresses in down markets. Earning di erences also in uence an agent s decision on whether to work as a broker. Figure 3 follows the same 1998 cohort and reports the fraction of agents who continue working as a broker for each quartile in each year. There are stark di erences in the exit rates among the four groups. Only 25% of the top quartile agents left by In contrast, about three quarters of the agents in the bottom quartile exited at some point during the ten-year period. Figure 3 presents our identi cation argument in a nutshell: di erences in the exit rate of agents earning di erent commissions allow us to identify their opportunity cost of being a broker. 8

9 3.2 Descriptive regressions We next investigate how competition a ects agent performance. The correlations we document here inform the modeling choices we make in the next section. In the regressions below, we measure the extent of competition by the number of competing real estate agents who work in the same market and year. We report estimates of agent performance, y imt, for agent i working in market m in year t from the following equation: y imt = m + t + s it + log(n mt ) + icmt : (1) where y imt is one of two measures of performance: commissions and the number of transactions. m and t are market and year xed e ects. s it is agent skill, proxied by the number of properties agent i intermediated the previous year following Table 3B. 7 The parameter of central interest is ; the coe cient of the competition measure, which reveals the impact of an increase in the number of competing agents on agent i s performance, adjusting for his skill. We estimate equation (1) for a xed cohort, de ned as the set of agents who are active or have entered as of a given initial year. This formulation avoids changes in the composition of agents, which would confound our results. For instance, the 1998 cohort analysis includes all agents active in Agents who entered in subsequent years are excluded from the regression, but they contribute to the competition variable log(n mt ). 8 Table 4A reports estimates of for the rst seven cohorts (the to the cohort). Estimates for later cohorts are much less precise as sample sizes drop. To allow for di erential competition e ects for established agents and others, we assign each agent in a cohort to four groups according to his commission in the initial cohort year, and estimate equation (1) separately for each group. Results of the commission regression and the number-of-transaction regression are reported in columns (1)-(5) and columns (6)-(10) of Table 4A, respectively. Since agent entry is cyclical and our competition measure increases during a booming market, we anticipate that our estimates are biased downward. Despite this, the estimates are signi cantly negative and sizable for almost all regressions we estimated, implying that incumbent agents receive lower commissions and conduct fewer transactions when competition intensi es. For example, a 10% increase in the level of competition is associated with a 1.1%-6.7% decrease in average commissions and a % reduction in the number of transactions, with the largest impact born by the 2004 cohort. Across agent quartiles, the estimates tend to be larger for the 2 nd and 3 rd quartiles, implying that competitors steal more business from middle-tier agents, and have a smaller impact on superstar agents. 9 Having documented a strong business stealing e ect, we now examine whether home sellers 7 We also estimated models using the number of years worked as a broker and found similar estimates. 8 Changes in the composition of cohorts do arise when agents exit. We also estimated equation (1) on the subset of agents who are active in all years (a balanced panel). These estimates produce larger estimates of the negative impacts of competition, although the di erences are not signi cant. 9 The coe cients for the bottom quartile agents are less precise since many have zero or one transaction in any given year. 9

10 bene t from more competition among agents. Let h imt be a measure of the sales experience (the likelihood of sale, days on the market, or sales price) of a home intermediated by agent i who works in market m in year t. We estimate a property level regression of the following form: h imt = m + t + s it + 0 X ht + log(n mt ) + imt : (2) where m ; t, s it ; and N mt are de ned as in equation (1). X ht represents a vector of property attributes including zip code xed e ects, the number of bedrooms, bathrooms, and other rooms, the number of garages, age, square footage, lot size, architectural style, whether it has a garden, type of heating, whether it is a condominium, a single family or a multi-family dwelling, and sometimes the list price. We report estimates of in Table 4B. When the number of competing agents in a market increases, the likelihood that a property sells decreases, even though more agents are typically associated with a booming market. The point estimate in column (1) implies that a 10% increase in the number of agents is associated with nearly a 1% reduction in the sales probability, whose sample average is 69%. It is possible that when there are more agents, sellers are more likely to list their property at a higher price to sh for a buyer. We add list price in column (2) to control for seller s preference. With this adjustment, the negative impact reduces to 0.6%, but still signi cant. Another possible explanation is that the composition of properties changes with the market condition: houses that are harder to sell (due to unobserved attributes) are more likely to be listed in a booming market. To examine this possibility, we interact the competition measure with indicators for before or after 2005 in column (3). The estimates for both periods remain negative, suggesting that the negative impact of competition is not solely driven by unobserved changes in the composition of properties listed in an up market. The impact of more agents on days on the market is negative, but insigni cant. It is possible that variation in days on the market are mostly driven by market-wide conditions, captured by market and year xed e ects in equation (2). Competition does seem to be associated with an increase in the sales price of a property, but the impact is modest when we control for the list price, as documented in columns (8) and (9). A 10% increase in competition generates a 0:14% increase in the sales price, which translates to about $600 for a typical home. The estimate is similar when we allow for di erential competition e ects before and after Since a higher sales price is a transfer from buyers to sellers and has a negligible impact on aggregate consumer surplus, in subsequent sections we do not focus on the impact agents have on sales prices. In summary, results from estimates of equation (2) suggest that the bene t home sellers receive from more agent competition is limited at best. 4 Econometric Model The patterns in the previous section show that there is a strong business stealing e ect and that competition from more agents does not improve agents quality of service as measured by sales 10

11 likelihood and time to sale. In this section, we incorporate competition among agents in modeling their entry and exit decisions. These decisions, together with observed commission revenue, allow us to estimate their opportunity cost of being a broker. We rst describe various elements of the model: the state variables, the revenue (or payo ) function, and the transition process of state variables. Then we present the Bellman equation and the value function and discuss some limitations of the model. 4.1 State variables To model the evolution of the housing market and how it a ects the entry and exit decisions of agents, we need to represent the housing market in terms of state variables. Since our data includes information on the attributes of each property that an agent intermediates, in principle, we could model how agents are matched to particular properties, and how this would impact their commission revenue. We do not pursue this rich representation and instead work with a more stylized version of the housing market. There are two main reasons for this simpli cation. First, including property-speci c features in the state space substantially increases its dimension. A large number of payo relevant state variables in dynamic models is challenging for estimation and in counterfactual analyses that involve solving for a new equilibrium. In particular, it is di cult to estimate the joint transition process of many state variables and compute a high-dimensional integral of an unknown value function. Second, even if it were possible to surmount these computational and estimation hurdles, our data do not include information on the characteristics of home sellers and buyers, making it formidable to model the matching process between households and agents without ad hoc assumptions. As a result, we choose a parsimonious representation of the housing market that still allows for a reasonable t of the main moments of the data. We assume that agents commissions are determined by two sets of (payo -relevant) variables: aggregate variables and their individual characteristics. The aggregate variables are the total number of houses listed on the market, the average housing price, the ratio of inventory over the number of properties sold in the previous year, and the number of competing agents. The total number of listed houses H mt counts all houses for sale as of the rst day of year t together with all new listings throughout the year in market m. The average house price P mt is the equal-weighted price of all houses that are sold in market m in year t. To construct the inventory-sales ratio, at the beginning of each month, we take the ratio of the number of listed properties in inventory (which includes new listings and unsold properties) and the number of properties sold in the previous year. Next, we average over 12 months to compute the inventory-sales ratio in year t, denoted by inv mt. This ratio is included as a proxy for market tightness, considered an important factor by the NAR who publishes a similar Market Tightness Index. In our application, this ratio is an important predictor of whether a listed property is sold and the amount of time it takes to sell. Individual characteristics include an agent s gender, rm a liation, the number of years they have worked as a broker, and a count of their past transactions. We assume that the rst three 11

12 aggregate state variables describing the housing market the number of houses for sale, the average price of houses, and the inventory-sales ratio transition exogenously. That is, we abstract away from potential feedback between agent entry and the aggregate housing market state variables since it seems unlikely that this accounts for a signi cant fraction of market-level variation. 4.2 Revenue function Realtors earn commissions either from sales (as listing agents) or purchases (as buyer s agents) of homes. We model these two components of agent revenue separately. Agent i s commissions from sales depends on his share of houses listed for sale and the probability that these listings are sold within the contract period. Since the aggregate variables are the same for all agents in market m and year t, the listing share only depends on individual and rival characteristics (we omit the market subscript m throughout this section). The following listing share equation can be derived from a static home seller s discrete choice model (presented in the appendix): ShL it = exp(xl it L + L it) Pk exp(xl kt L + L (3) kt ): The variables Xit L include agent i s demographics, work experience, rm a liation, and proxies for agent skill. The variable L it represents his unobserved quality (observed by all agents, but unobserved by the econometrician), much like unobserved quality variables in Berry, Levinsohn, and Pakes (1995) and other discrete choice models. We report estimates assuming that L it is independent across periods. In Section 6.1, we present evidence that correlated unobserved state variables may not be important once we include our proxy for agent skill. This assumption is also needed because of computational di culties of incorporating correlated state variables in dynamic discrete choice models. The denominator in equation (3) L t X k exp(x L kt L + kt ); (4) is sometimes called the inclusive value (see, e.g., Aguirregabiria and Nevo (2010)). It measures the level of competition agents face in obtaining listings. Rather than tracking all rivals decisions, they behave optimally against the aggregate competition intensity L t : This approximation of competition can be motivated by the large number of agents per market. Agents only receive commissions when listings are sold. The probability that agent i s listings are sold is assumed to have the following form: Pr(sell it ) = exp(xs it S ) 1 + exp(x S it S ) ; where Xit S includes measures of aggregate housing market conditions (total number of houses listed, the inventory-sales ratio, etc.), as well as his own characteristics. Since we treat the sales price as 12

13 exogenous, this formulation does not allow for a trade-o between the probability of sale and the sales price. An agent s total commission from selling listed houses is: R Sell it = r H t P t ShL it Pr(sell it ); where r is the commission rate, H t is the aggregate number of houses listed, and P t is the average price index. We develop the model for an agent s commissions from representing buyers in a similar way: R Buy it = r H B t P t ShB it ; where Ht B is the total number of houses bought by all home buyers, P t is the same as before, and ShB it is agent i s share of the buying market: ShB it = exp(xb it B + B it) Pk exp(xb kt B + B (5) kt ): Here, X B it and B it are his observed and unobserved characteristics, respectively. Similar to the listing share, the inclusive value on the buyer side is: B t X k exp(x B kt B + B kt ): To reduce the number of state variables, we make the simplifying assumption that H B t = 0:69H t ; where 0:69 is the average probability that houses are sold. In our sample the correlation between H B t and H t is 0.94, so this simpli cation allows us to reduce the state space. Since an agent s revenue depends on H t P t, we group these two variables together as HP t, a single state variable that measures the aggregate size (in dollars) of a housing market. Finally, agents earn commissions as both buyer s and seller s agents. As a result, agent i s earnings at any given set of payo -relevant state variables S it = fx L it ; XS it ; XB it ; HP t; inv t ; L t ; B t g is: R(S it ) = R Sell (S it ) + R Buy (S it ) = 0:015 HP t (ShL it Pr Sell it +ShB it 0:69): Despite this stylized representation of the housing market, the correlation between the model s predicted revenue and the observed revenue is The model also captures well the upward and downward trend of observed revenues. We provide details on the model s tness in Section Transition process of state variables When agents consider entry and exit, they factor in both their current revenue and their future prospects as realtors, which are determined by the exogenous state variables as well as rival agents 13

14 entry and exit decisions. Table 2A shows that entry nearly doubled in 2005 and then dropped substantially afterward. We do not explicitly model agent s beliefs on how the aggregate state variables evolve. Instead, we adopt a standard AR(1) model with trend breaks before and after 2005, when house prices peaked in our sample. The aggregate state variables are assumed to evolve according to the following equation: S mt+1 = T 0 1 [t < 2005] + T 1 1 [t 2005] + T 2 S mt + m + mt ; (6) where T 0 and T 1 are coe cients of the trend break dummies, 1fg are indicator functions, T 2 is a matrix of autoregressive coe cients, m is the market xed e ect, and mt is a mean-zero multivariate normal random variable. Market xed e ects in equation (6) are included to control for size di erences across markets: the largest 5 markets have twice as many listings as the smallest 5 markets. In JP2, we show that omitting market xed e ects leads to biased estimates for the autoregressive coe cients. We also investigated splitting the sample at year 2005 and estimating a separate transition process for each sub-sample without much success. The R 2 for the second part of the sample is very low, since we have only a few periods per market after An alternative to the structural break is to add multiple lags and high-order polynomials. We prefer equation (6) given that its R 2 is high (ranging from 0.77 to 0.96) and that our panel is relatively short. An agent s skill is also modeled as an AR(1) process, including the trend break as above. 4.4 Entry and exit decisions In the econometric model, agents can make career adjustments each period: some incumbent agents continue to work as realtors, others leave it (exit), and new agents become brokers (entry). At the beginning of a period, agents observe the exogenous state variables, their own characteristics, as well as two endogenous variables L t 1 and B t 1 at the end of the previous period. L t and B t are measures of the competition intensity and are determined by all agents entry and exit decisions jointly: they increase when more people become realtors and decrease when realtors quit and seek alternative careers. Agents also observe their private idiosyncratic income shocks and simultaneously make entry and exit decisions. Since agents can start earning income as soon as they nd clients, we assume that there is no time lag between entry (becoming an agent) and earning commissions. This assumption contrasts with the literature on the dynamic entry and exit decisions of rms, which assume that rms pay an entry cost at period t and start generating revenues in period t + 1 after a one-period delay due to installing capital and building plants (see, e.g., Ericson and Pakes (1995)). Let Z denote exogenous state variables and individual characteristics and Y denote endogenous state variables L and B. The Bellman equation for an active agent i is: ~V (Z it ; Y t 1 ) = E ~" max ( E [R(Z it ; Y t )jz it ; Y t 1 ] c + ~" 1it + E ~ V (Z i;t+1 ; Y t jz it ; Y t 1 ) ~" 0it (7) 14

15 where E [R(Z it ; Y t )jz it ; Y t 1 ] is his expected commission revenue conditional on observed state variables. Conditioning on state variables, the revenue function also depends on L it and B it which we integrate out using their empirical distributions. 10 Since income shocks are private, agent i does not observe Y t that is determined by all rivals entry and exit at period t. Instead, he forms an expectation of his commission revenue for the coming period if he continues working as a broker. The variable c captures agent i s costs of brokering house transactions. It includes his foregone labor income from working in an alternative profession, as well as the per-period xed cost of being an agent due to the expense of renting o ce space, the cost of maintaining an active license, and resources devoted to building and sustaining a customer network. We assume that the cost of being a broker does not depend on the number of houses he handles, because the marginal monetary cost of listing more properties is likely swamped by the xed costs. We report counterfactual results under di erent assumptions on marginal cost as a robustness check. In all speci cations we consider, c di ers across markets, but is the same for agents within a market. In the main speci cation, c is xed throughout the sample period, but we also present results allowing it to vary over time. The econometric model treats exit as a terminating action. Re-entering agents account for about 9% of our sample. Relaxing this assumption would require estimating two value functions and substantially increase the complexity of the model. 11 Private shocks ~" 0 and ~" 1 are assumed to be i.i.d. extreme value random variables with standard deviation 1 ; where 1 1 > 0. Denoting the expected commission revenue E [R(Z it ; Y t )jz it ; Y t 1 ] as R(Z it ; Y t 1 ), and multiplying both sides of equation (7) by 1, the original Bellman equation can be rewritten as: ( V (Z it ; Y t 1 ) = E " max 1R(Zit ; Y t 1 ) 1 c + " i1t + EV (Z it+1 ; Y t jz it ; Y t 1 ) " i0t ; where V (Z it ; Y t 1 ) = 1 ~ V (Zit ; Y t 1 ) and " ikt = 1 ~" ikt ; for k = 0; 1: Given the distributional assumptions on "; the Bellman equation is simpli ed to the usual log-sum form: V (Z it ; Y t 1 ) = log 1 + exp R(Z it ; Y t 1 ; ) + EV (Z it+1 ; Y t jz it ; Y t 1 ) ; where we have replaced 1 R(Zit ; Y t 1 ) 1 c with R(Z it ; Y t 1 ; ) to keep the notation simple. The main focus of the empirical exercise is estimating = f 1 ; 2 g; with 2 = 1 c. The probability that incumbent agent i is active at the end of period t is: Pr(stay it jz it ; Y t 1 ; ) = exp R(Z it ; Y t 1 ; ) + EV (Z it+1 ; Y t jz it ; Y t 1 ) 1 + exp R(Z : (8) it ; Y t 1 ; ) + EV (Z it+1 ; Y t jz it ; Y t 1 ) 10 We ignore the dependence of L t and B t on it, which we suspect is negligible given the large number of agents included in L and B: 11 The estimation strategy would be similar, except that we need to use the exit choice probability to recast one of the choice-speci c value functions as a xed point of a Bellman equation following Bajari, Chernozhukov, Hong, and Nekipelov (2009). 15

16 Let W it = 1 be an indicator that agent i remains active at t. The log likelihood (for incumbent agents) is: LL()= X i;t 1 [W it = 0] log[1 Pr(stay it j)]+ X i;t 1 [W it = 1] log[pr(stay it j)]: (9) Provided we are able to solve for EV and calculate the choice probability Pr(stay it j); we can estimate by maximizing equation (9). Potential entrants must pay a fee (entry cost) to become a broker. They enter if the net present value of being an agent is greater than the entry cost (up to some random shock). The Bellman equation for potential entrant j is: ( V E (Z jt ; Y t 1 ) = E " max + R(Z jt ; Y t 1 ; ) + " j1t + EV (Z jt+1 ; Y t jz jt ; Y t 1 ) " j0t = log 1 + exp + R(Z jt ; Y t 1 ; ) + EV (Z jt+1 ; Y t jz jt ; Y t 1 ) : Just as in equation (8), the probability of entry is: Pr(entry jt jz jt ; Y t 1 ; ; ) = exp + R(Z jt ; Y t 1 ; ) + EV (Z jt+1 ; Y t jz jt ; Y t 1 ) 1 + exp + R(Z jt ; Y t 1 ; ) + EV (Z jt+1 ; Y t jz jt ; Y t 1 ) : Let E jt = 1 be an indicator that potential entrant j enters at t. The log likelihood of observing N E t = j E jt new entrants out of a maximum of N E potential entrants is: LL E () = X j N E ;t 1[E jt = 1]log[Pr(entry jt j; )]+ X j N E ;t 1[E jt = 0]log[1 Pr(entry jt j; )]: (10) Since the entry cost estimate ^ is sensitive to the assumption of the maximum number of potential entrants N E ; we estimate equation (10) separately from the main model (9). We report estimates of entry costs under three di erent assumptions on N E. 4.5 Discussion on modeling assumptions The main structural parameter of interest is c, the average agent s total costs of brokering transactions. The current formulation of the model does not allow c to depend on state variables. This is constrained by the fact that we only observe one action for each active agent (stay or exit) and cannot separately identify the impact of a state variable working through c versus its impact working through revenue R on agent actions. Likewise, we cannot allow for a variable cost component in c (which depends on the number of transactions) because agent revenue is proportional to his total number of transactions. However, the model does allow c to vary across markets, as might be expected if outside opportunities are related to market conditions. Some real estate brokers work part time. According to NAR (2007), 79% of realtors report that real estate brokerage is their only source of income. For the other 21% of agents holding more 16

17 than one job, we do not observe their income from other sources. However, our estimate ^c is the relevant measure of agents time devoted to working as brokers. Suppose an agent has two jobs, earning $35,000 as a broker and $10,000 from a second job. If we observe him exit after earning $30,000, then his opportunity cost of working as a broker is between $30,000 and $35,000 (ignoring the optional value of future earnings), even though the value of his total working time is higher. Having part-time agents reduces ^c, but this correctly measures the average value of time that agents devoted to being a broker. We attempted to formally address part-time agents using a discrete mixture model that allows two types of agents with di erent opportunity costs, but the likelihood was at in a large region of parameter values. Two other features of the model are also due to data constraints that we do not observe all aspects of the interaction between home-sellers, agents, and the housing market. First, the model is silent on possible unobserved bene ts that consumers derive from competition among realtors. For instance, we do not measure gifts that agents give away in marketing their services. In addition, when there is enhanced competition from entry, agents might work harder to satisfy requests from their clients and provide better services. The results in Section 3 suggest that these possibilities did not translate into gains for sellers on the likelihood of sale and days on the market. It is possible, however, that buyers bene t from the variety a orded by a large pool of agents. If this is an important source of consumer surplus, then our counterfactual results miss the losses to consumers with fewer agents. Second, since we do not observe the actual contract terms between agents and their rms, we assume that agents keep 60% of total commissions, based on the 2007 national survey conducted by NAR. This assumption a ects our estimate proportionately: if the average commission is underestimated by %, then 1 will be over-estimated by the same amount. As a result, the opportunity cost c = 2 1 will be under-estimated by %. 5 Solution Method As explained above, the unknown value function V () is implicitly de ned by a functional Bellman equation. The ability to quickly compute the value function is a crucial factor in most empirical dynamic models and in many cases is a determining factor in model speci cation. In this section, we describe our solution algorithm. JP2 contains additional computational details and Monte-Carlo results. To simplify notation, we omit subscripts throughout this section, and use S to denote the vector of state variables. 5.1 Di culties with existing approaches We began our analysis with the traditional approach of discretizing the state space, but met with substantial memory and computational di culties when we tested our model with four state variables. One of the challenges involves calculating the future value, EV (S 0 js), a high-dimensional integral of an unknown function. The quadrature rules require evaluating the value function V (S) 17

18 at quadrature points that do not overlap with grid points. Since V (S) is unknown at any point outside grids, we need to interpolate V (S) from grid points to quadrature points. With four state variables and ten grids each, more than 95% of our computing time was spent on interpolation. As a result, solving the value function using the Bellman iteration V k (S) = (S; E(V k 1 (S 0 js)) for a given parameter value was slow and often took a couple of hours. Moreover, the memory requirements of discretization increases exponentially. 12 Another factor that discouraged us from discretization is that there are far fewer data points than the size of the state space when the number of state variables is large. Discretizing the state space and solving the value function for the entire state space implies that most of the time in estimation is spent solving value function V (S) for states that are never observed in the data (and hence not directly used in the estimation). In addition, both discretization and interpolation introduce approximation errors that grow with the number of state variables. The alternative method we pursue approximates the value function V (S) using sieves where unknown functions are approximated by parametric basis functions (see, e.g., Chen (2007)). For our application, this approach has several bene ts. First, the sieve approximation eliminates the need to iterate on the Bellman equation to solve the value function, and therefore avoids the most computationally intensive part of estimation. The Bellman equation is instead cast as a constraint of the model that has to be satis ed at the parameter estimates. This formulation reduces the computational burden and makes it feasible to solve for the equilibrium of models with medium to high dimensions. In addition, the algorithm does not spend time calculating the value function in regions of the state space not observed in the sample. The method has the potential to improve upon methods that require calculating the value function for the entire state space, whose number of elements is often an order of magnitude larger than a typical sample size. For example, with six state variables (which is the number of state variables in our base speci cation) and ten grids for each, there are 10 6 elements in the state space. There are two main downsides of our approach: a) nite-sample biases from the approximation and b) the non-parametric approximation converges to the true value function at a rate slower than the square root of the sample size. JP2 documents Monte-Carlo evidence that the method works well in our application: with a reasonable number of basis functions, the value function approximation error is small, the bias in parameter estimates is negligible, and the computation is very fast. We now present the estimation procedure in detail. 5.2 Sieve estimation of the value function Recall that our Bellman equation is: V (S) = log 1 + exp R(S; ) + EV (S 0 js) : (11) 12 We ran out of memory on a server with 32GB of RAM when we experimented with 20 grid points for each of the four state variables. 18

19 Kumar and Sloan (1987) show that if the Bellman operator is continuous and EV (S 0 js) is nite, then sieve approximation approaches the true value function arbitrarily close as the number of sieve terms increases. 13 approximate for the value function V (S): This fact provides the theoretical foundation for using basis terms to Speci cally, let V (S) be approximated by a series of J basis functions u j (S): V (S) ' JX b j u j (S); (12) j=1 with unknown coe cients fb j g. Substitution of equation (12) into equation (11) leads to a nonlinear equation: JX JX b j u j (S) = + exp 4 R(S; ) + b j Eu j (S 0 js) 5A : j=1 This equation should hold at all states observed in the data. Our approach is to choose fb j g to best- t this non-linear equation in least-squared-residuals : o JX n^bj = arg min fb j g b j u j (S (k) ) j=1 0 2 j=1 + exp 4 R(S(k) ; ) + 31 JX b j Eu j (S 0 js (k) ) 5A (13) 2 where K S (k) k=1 denotes state values observed in the data, and kk 2 is the L2 norm. Essentially, fb j g are solutions to a system of rst order conditions that characterize how changes in fb j g a ect violations of the Bellman equation. There are many possible candidates for suitable basis functions u j (S) including power series, Fourier series, splines, and neural networks. Jia and Genz (2011) compare a group of popular basis functions using Monte-Carlo simulations. In general, the best basis function is application speci c, but well-chosen basis functions should approximate the shape of the value function. A large number of poor basis functions can create various computational problems and estimation issues such as large bias and variance. Since we observe agents revenue directly, we exploit information embodied in the revenue function to guide our approximation of the value function. In general, if the revenue function R(S) increases in S and the transition process T S also increases in S, then the value function V (S) increases in S: 14 This property suggests the following approach: use basis functions that t the revenue function R(S) to approximate the value function in the Bellman equation. Since these basis functions are chosen to preserve the shape of R(S); they should also capture the shape of the value function. Choosing basis terms in high-dimensional models is not a simple matter; hence, we want an adaptable procedure to economize on the number of terms to reduce numerical errors and parameter 13 We thank Alan Genz for suggesting this reference. 14 The formal proof of this fact follows from the Contraction Mapping Theorem and is contained in the appendix. In our application, this observation applies since R() increases in L and B; and L 0 and B 0 increase in L and B (i.e., the transition process increases in L and B). j=1 19

20 variance. We adopt the Multivariate Adaptive Regression Spline (MARS) method popularized by Friedman (1991) and Friedman (1993) to nd spline terms that approximate the revenue function to a desired degree. MARS repeatedly splits the state space along each dimension, adds spline terms that improve the tness according to some criterion function, and stops when the marginal improvement of the t is below a threshold (1:0 10 3, for example). 15 Once we obtain a set of spline basis terms f^u j (S)g that best t our revenue function R(S), we substitute them for f^u j (S)g in equation (13). The estimated coe cients f^b j g are those that minimize the squared di erence between the left-hand-side and right-hand-side of the Bellman equation, where the value function is approximated by P J j=1 ^b j u j (S (k) ) for each point in the state space S (k). As in other applications of Mathematical Programming with Equilibrium Constraints (see, e.g., Judd and Su (2008) and Dube, Fox, and Su (2009)), we impose equation (13) as a constraint and do not explicitly solve for f^b j g in each iteration of the estimation procedure. The number of spline terms J is an important component of estimation. We propose a data driven method to determine J. Let ^ J denote the parameter estimates when the value function is approximated by J spline terms. We increase J until parameter estimates converge, when the element by element di erence between ^ J and ^ J 1 is smaller than half of its standard deviation (which we estimate using non-parametric bootstrap simulations). 5.3 Identi cation Identi cation of 1 and 2 follows from the identi cation argument of a standard entry model. Substantial exit following a moderate reduction in revenue implies a relatively large value of 1, the coe cient which measures sensitivity to revenue. On the other hand, if exit does not vary much with reductions in revenue, then 1 is small. The coe cient 2 is identi ed from the level of revenue at which exits start to occur. Identi cation of the spline coe cients b follows from Hotz and Miller (1993), which proved that di erences in choice-speci c value functions can be identi ed from observed choice probabilities. In our application, the value function associated with the outside option is set to 0. With this normalization, choice probabilities directly lead to identi cation of the value function and the spline coe cients b. 6 Estimates We rst examine estimates of the revenue function and state variables transition process, then present opportunity cost estimates and discuss the model s t. Throughout this section, we bring back the market subscript m. Following Hajivassiliou (2000), we standardize all state variables to avoid computer over ow errors. The aggregate state variables, HP mt ; inv; L mt ; and B mt are standardized with zero mean and 1 standard deviation; the skill variable s it is standardized with 15 We use the R package earth (which implements MARS, written by Stephen Milborrow), together with the L 2 norm as our criterion function. We search for spline knots and spline coe cients that minimize the sum of the square of the di erence between the observed revenue and the tted revenue at each data point. 20